Solve for $x$ : $ 8|x - 6| + 4 = -5|x - 6| + 9 $
Solution: Add $ {5|x - 6|} $ to both sides: $ \begin{eqnarray} 8|x - 6| + 4 &=& -5|x - 6| + 9 \\ \\ { + 5|x - 6|} && { + 5|x - 6|} \\ \\ 13|x - 6| + 4 &=& 9 \end{eqnarray} $ Subtract ${4}$ from both sides: $ \begin{eqnarray} 13|x - 6| + 4 &=& 9 \\ \\ { - 4} &=& { - 4} \\ \\ 13|x - 6| &=& 5 \end{eqnarray} $ Divide both sides by ${13}$ $ \dfrac{13|x - 6|} {{13}} = \dfrac{5} {{13}} $ Simplify: $ |x - 6| = \dfrac{5}{13}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 6 = -\dfrac{5}{13} $ or $ x - 6 = \dfrac{5}{13} $ Solve for the solution where $x - 6$ is negative: $ x - 6 = -\dfrac{5}{13} $ Add ${6}$ to both sides: $ \begin{eqnarray} x - 6 &=& -\dfrac{5}{13} \\ \\ {+ 6} && {+ 6} \\ \\ x &=& -\dfrac{5}{13} + 6 \end{eqnarray} $ Change the ${ + 6}$ to an equivalent fraction with a denominator of $13$ $ x = - \dfrac{5}{13} {+ \dfrac{78}{13}} $ $ x = \dfrac{73}{13} $ Then calculate the solution where $x - 6$ is positive: $ x - 6 = \dfrac{5}{13} $ Add ${6}$ to both sides: $ \begin{eqnarray} x - 6 &=& \dfrac{5}{13} \\ \\ {+ 6} && {+ 6} \\ \\ x &=& \dfrac{5}{13} + 6 \end{eqnarray} $ Change the ${ + 6}$ to an equivalent fraction with a denominator of $13$ $ x = \dfrac{5}{13} {+ \dfrac{78}{13}} $ $ x = \dfrac{83}{13} $ Thus, the correct answer is $x = \dfrac{73}{13} $ or $x = \dfrac{83}{13} $.